Auburn Algebra Seminar
Fall 2024
Schedules from past semestersSeminars will be held in Parker 358 on Tuesdays from 2:30 to 3:20.
Schedule:
August 20: Ritvik Ramkumar (Cornell). This talk will be on Zoom: use this link.Title: Hilbert scheme of points on threefolds
Abstract: The Hilbert scheme of d points on a smooth variety X, denoted by Hilb^d(X), is an important moduli space with connections to various fields, including combinatorics, enumerative geometry, and complexity theory, to name a few. In this talk, I will introduce this object and review some well-known results when X is a curve or a surface. The main focus of this talk will be on the case when X is a threefold. Specifically, I will present some tantalizing open questions and delve into describing the structure of the smooth points on Hilb^d(X). If time permits, I will also discuss the mildly singular points of this Hilbert scheme. This is all joint work with Joachim Jelisiejew and Alessio Sammartano.
August 27: Austin Conner (Harvard)
Title: Geometry and the complexity of matrix multiplication
Abstract: In 1968, Strassen discovered a faster way to multiply matrices than the standard row-column multiplication. Since then a fundamental question in theoretical computer science has been to determine just how fast matrices may be multiplied. Strassen's algorithm is at core a certain method to multiply 2 by 2 matrices using 7 multiplications. The data describing this method is equivalently an expression to write the structure tensor of the 2 by 2 matrix algebra as a sum of 7 decomposable tensors. Any such decomposition of an n by n matrix algebra yields a Strassen type algorithm, and Strassen showed that one essentially cannot do better than algorithms coming from such decompositions. Bini later showed all of the above remains true when we allow the decomposition to depend on a parameter and take limits. I discuss a technique for lower bounds for this decomposition problem, border apolarity. Two key ideas to this technique are (i) to not just look at the sequence of decompositions, but the sequence of ideals of the point sets determining the decompositions and (ii) to exploit the symmetry of the tensor of interest to insist that the limiting ideal has an extremely restrictive structure. I discuss its applications to the matrix multiplication tensor and other tensors potentially useful for obtaining upper bounds via Strassen's laser method.
September 3: Ayah Almousa (University of South Carolina)
Title: Stirling numbers and Koszul algebras with symmetry
Abstract: Stirling numbers c(n,k) and S(n,k) of the first and second kind are the answers to two counting problems: how many permutations of n letters have k cycles, and how many set partitions of [n] have k blocks? The c(n,k) also give the Hilbert function for certain well-studied Koszul algebras with symmetry: the cohomology of configurations of n distinct labeled points in $d$-space, also known as the Orlik-Solomon algebras and graded Varchenko-Gelfand algebras for type A reflection hyperplane arrangements. We discuss how the S(n,k) give the Hilbert series for their less-studied Koszul dual algebras. This includes relating the symmetric group action on the original algebras and on their Koszul duals, representation stability in the sense of Church and Farb, and branching rules that lift Stirling number recursions. This is joint work with Vic Reiner and Sheila Sundaram.
September 10: Sean Grate (Auburn)
Title: Betti numbers of connected sums of graded Artinian Gorenstein algebras
Abstract: Considered as an algebraic analog for the connected sum construction from topology, the connected sum construction introduced by Ananthnarayan, Avramov, and Moore is a method to produce Gorenstein rings. Joint with Nasrin Altafi, Roberta Di Gennaro, Federico Galetto, Rosa M. Miró-Roig, Uwe Nagel, Alexandra Seceleanu, and Junzo Watanabe, we determine the graded Betti numbers for connected sums and fiber products of Artinian Gorenstein algebras, where the fiber product in the local setting was obtained by Geller. We also show that the connected sum of doublings is the doubling of a fiber product ring. I will discuss these results through some examples and Macaulay2 code.
September 17: Colin Crowley (University of Oregon)
Title: Orlik-Terao algebras and internal zonotopal algebras
Abstract: In 2017 Moseley, Proudfoot, and Young conjectured that the reduced Orlik-Terao algebra of the braid matroid was isomorphic as a symmetric group representation to the cohomology of a certain configuration space. This was proved by Pagaria in 2022. We generalize Pagaria's result from the braid arrangement to arbitrary hyperplane arrangements and recover a new proof in the case of the braid arrangement. Along the way, we give formulas for several other invariants of a hyperplane arrangement. Joint with Nick Proudfoot.
September 24: Michael Brown (Auburn)
Title: Noncommutative sheaf cohomology
Abstract: I will give a brief introduction to noncommutative projective geometry, with a view toward a pair of projects concerning noncommutative sheaf cohomology (one joint with Daniel Erman and Greg Smith, and another joint with Prashanth Sridhar).
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October 8: Hal Schenck (Auburn)
Title: Syzygies of permanental ideals
Abstract: We describe the minimal free resolution of the ideal of 2×2 subpermanents of a 2×n generic matrix M. In contrast to the case of 2×2 determinants, the 2×2 permanents define an ideal which is neither prime nor Cohen-Macaulay. We combine work of Laubenbacher-Swanson on the Gröbner basis of an ideal of 2×2 permanents of a generic matrix with our previous work connecting the initial ideal of 2×2 permanents to a simplicial complex. The main technical tool is a spectral sequence arising from the Bernstein-Gelfand-Gelfand correspondence. Joint work with F. Gesmundo, H. Huang, J. Weyman.
October 15: Justin Lyle (Auburn). Note: this talk begins at 2:00.
Title: Homological Aspects of Commutative Rings
Abstract: Homological methods have been integral to our understanding of both commutative and noncommutative algebras since the so-called "Homological Invasion" of the 1950's, and have led to solutions of numerous problems not intrinsically homological in nature. Despite this, there are several fundamental aspects that remain mysterious, with several problems from the 1950's-1970's remaining stubbornly open, including the Finitistic Dimension Conjecture, the Nakayama Conjecture and its generalized variant, the Tachikawa Conjectures, and the Auslander-Reiten Conjecture. In this talk, we give an overview of the historical and broader contexts of these problems, and discuss recent progress of the speaker on some related questions, as a consequence leading to new broad cases of the Auslander-Reiten conjecture.
October 22: Ian Tan (Auburn)
Title: Four-qubit critical states
Abstract: Let H be the Hilbert space of unnormalized four-qubit state vectors. By a result of Verstraete, Dehaene, and De Moor, ||f||^(1/m) is an entanglement monotone for any complex homogeneous polynomial f on H of degree m > 0 invariant under the action of the SLOCC group. We observe that many highly entangled or useful four-qubit states that appear in prior literature are stationary points of ||f|| for natural choices of f . This motivates the search for more stationary points. Using the notion of critical points (in the sense of the Kempf-Ness theorem) together with results from Vinberg’s theory of theta groups, we reduce the complexity of the problem significantly. After reduction, we solve systems of polynomial equations with techniques from numerical algebraic geometry, recovering the stationary points known to us at the beginning and more.
October 29:
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November 5:
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November 12: Prashanth Sridhar (Auburn)
Title: Noncommutative geometry over dg-algebras
Abstract: The general motto of noncommutative algebraic geometry is that any category sufficiently close to the derived category of a variety should be regarded as a noncommutative variety. Using this principle, pioneering work of Artin-Tate-Van den Bergh-Zhang extends important aspects of projective geometry to the noncommutative setting. I’ll talk about extensions of this theory to noncommutative spaces associated to dg-algebras with a focus on how it feeds back into the commutative setting. In particular, I’ll discuss a generalization of a celebrated theorem of Orlov concerning the derived category of a projective complete intersection. Joint work with Michael K. Brown.
November 19: John Cobb (Auburn)
Title: Multigraded Syzygies
Abstract: I’ll describe two results that fit a recent theme of research in commutative algebra and algebraic geometry: pushing known results from the graded to the multigraded setting. One involves using logic to get a bound on projective dimension (Stillman’s Conjecture), and the other using homological techniques to get a bound on the regularity of curves. In the pretalk, I’ll introduce the main objects and theirs motivation more thoroughly.
December 3: Jose Franco (University of North Florida). This talk will be on Zoom: use this link.
Title: A theory of alternative means of matrices
Abstract: In this talk, we introduce a theory of alternative means of positive definite matrices. We define the alternative mean associated to a normalized operator monotone function by $$A\hat{\sigma}_f B = f(A^{-1}\sharp B)Af(A^{-1}\sharp B),$$ for positive definite matrices $A$ and $B$. We study the adjoint, transpose, and duals of these means and obtain a characterization of the spectral geometric means in terms of alternative means and their duals. Additionally, we show the deep connection between the alternative means and the near order of matrices. In particular, we obtain in-betweenness results of the type $A\preceq A\hat{\sigma}_f B \preceq B.$